Streams, graphs and global attractors of dynamical systems on locally compact spaces

In a recent article, we introduced the concept of streams and graphs of a semiflow. An important related concept is the one of semiflow with compact dynamics, which we defined as a semiflow $ F $ with a compact global trapping region. In this follow-up, we restrict to the important case where the phase space $ X $ is locally compact and we move the focus on the concept of global attractor, a maximal compact set that attracts every compact subset of $ X $. A semiflow $ F $ can have many global trapping regions but, if it has a global attractor, this is unique. We modify here our original definition and we say that $ F $ has compact dynamics if it has a global attractor. We show that most of the qualitative properties of $ F $ are inherited by the restriction of $ F $ to its global attractor and that, in case of Conley's chains stream of $ F $, the qualitative behavior of $ F $ and of its restriction to its global attractor coincide. Moreover, if $ F $ is a continuous-time semiflow, then its graph is identical to the graph of its time-1 map. Our main result is that, for each semiflow $ F $ with compact dynamics over a locally compact space, the graphs of the prolongational relation of $ F $ and of every stream of $ F $ are connected if the global attractor is connected.